Eric made two investments: Investment Q has a value of $500 at the end of the first year and increases by $45 per year. Investment R has a value of $400 at the end of the first year and increases by 10% per year. Eric checks the value of his investments once a year, at the end of the year. What is the first year in which Eric sees that investment R's value exceeded investment Q's value?

Question

Eric made two investments:  Investment Q has a value of  $500 at the end of the first year and increases by  $45 per year. Investment R has a value of  $400 at the end of the first year and increases by  10% per year.  Eric checks the value of his investments once a year, at the end of the year. What is the first year in which Eric sees that investment R's value exceeded investment Q's value?

Bytelearn · Accepted Answer

Define Q(n): Step 1: Define the value of investment Q after n years. Investment Q starts at $500 and increases by $45 each year. So, the value of investment Q after n years can be represented as: Q(n) = 500 + 45nDefine R(n): Step 2: Define the value of investment R after n years. Investment R starts at $400 and increases by 10% each year. The value of investment R after n years can be represented as: R(n) = 400 * (1 + 0.10)^nDetermine year when R(n) exceeds Q(n): Step 3: Determine the year when R(n) exceeds Q(n). We need to find the smallest integer n for which R(n) > Q(n). This means we are looking for the smallest n such that: 400 * (1.10)^n > 500 + 45nSolve inequality: Step 4: Solve the inequality by trial and error or algebraically. Since the inequality involves an exponential term and a linear term, it's not straightforward to solve algebraically. We will use trial and error, starting with n = 1 and increasing n until R(n) exceeds Q(n).Calculate Q(n) and R(n) for n=1: Step 5: Calculate the value of Q(n) and R(n) for successive years. For n = 1: Q(1) = 500 + 45(1) = 545 R(1) = 400 * (1.10)^1 = 440 R(1) is not greater than Q(1).Increment n and calculate: Step 6: Increment n and calculate again. For n = 2: Q(2) = 500 + 45(2) = 590 R(2) = 400 * (1.10)^2 = 484 R(2) is not greater than Q(2).Continue incrementing n: Step 7: Continue incrementing n. For n = 3: Q(3) = 500 + 45(3) = 635 R(3) = 400 * (1.10)^3 = 532.4 R(3) is not greater than Q(3).Continue incrementing n: Step 8: Continue incrementing n. For n = 4: Q(4) = 500 + 45(4) = 680 R(4) = 400 * (1.10)^4 = 585.64 R(4) is not greater than Q(4).Continue incrementing n: Step 9: Continue incrementing n. For n = 5: Q(5) = 500 + 45(5) = 725 R(5) = 400 * (1.10)^5 = 644.204 R(5) is not greater than Q(5).Continue incrementing n: Step 10: Continue incrementing n. For n = 6: Q(6) = 500 + 45(6) = 770 R(6) = 400 * (1.10)^6 = 708.624 R(6) is not greater than Q(6).Continue incrementing n: Step 11: Continue incrementing n. For n = 7: Q(7) = 500 + 45(7) = 815 R(7) = 400 * (1.10)^7 = 779.4864 R(7) is not greater than Q(7).Continue incrementing n: Step 12: Continue incrementing n. For n = 8: Q(8) = 500 + 45(8) = 860 R(8) = 400 * (1.10)^8 = 857.43504 R(8) is not greater than Q(8).Continue incrementing n: Step 13: Continue incrementing n. For n = 9: Q(9) = 500 + 45(9) = 905 R(9) = 400 * (1.10)^9 = 943.178544 R(9) is greater than Q(9). Therefore, the first year in which investment R's value exceeded investment Q's value is after 9 years.

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